Sam Tyner @sctyner
@sctyner
Interactions in animation aren’t realistic - people don’t behave like bacteria in petri dishes!
Can we use network modeling to answer these questions?
Another public health concern - underage drinking & smoking (Steglich, Snijders & Pearson 2010)
Data from “Teenage Friends and Lifestyle Study” (Pearson & Mitchell (2000); Pearson & West (2003))
Just consider modeling network change for now
Rate at which actors are selected to act
When actor \(i\) can change, it tries to maximize its objective function:
\[f_i(x, \mathbf{z}, \boldsymbol{\beta}) = \sum_k \beta_k s_{ik}(x, \mathbf{z})\]
The probability that actor \(i\) will change the tie to actor \(j\) is:
\[p_{ij} = \frac{\exp\left\{f_i(x(i\leadsto j), \mathbf{z}, \boldsymbol{\beta})\right\}}{\sum_h \exp\left\{f_i(x(i\leadsto h), \mathbf{z}, \boldsymbol{\beta})\right\}}\] - \(x(i\leadsto j)\) is the network identical to the current state, \(x\), except for \(x_{ij}\), which becomes \(1-x_{ij}\)
| Network Effect | Sufficient Statistic | Interpretation |
|---|---|---|
| outdegree* | \(s_{i1}(x) = \sum_j x_{ij}\) | Popularity |
| reciprocity* | \(s_{i2}(x) = \sum_j x_{ij}x_{ji}\) | Reciprocated relationships |
| transitive triplets | \(s_{i3}(x) = \sum_{j,h} x_{ij}x_{jh}x_{ih}\) | Your friend becomes my friend |
| Covariate Effect | Sufficient Statistic | Interpretation |
|---|---|---|
| covariate-alter | \(s_{i4}(x) = \sum_j x_{ij}z_j\) | Effect of my friend’s behavior on friendship |
| covariate-ego | \(s_{i5}(x) = z_i\sum_j x_{ij}\) | Effect of my behavior on friendship |
| same covariate | \(s_{i6}(x) = \sum_j x_{ij} \mathbb{I}(z_i = z_j)\) | Birds of a feather flock together |
| Network Effect | Interpretation | Visualization |
|---|---|---|
| outdegree* | Popularity | |
| reciprocity* | Reciprocated relationships | |
| transitive triplets | Your friend becomes my friend |
| Covariate Effect | Interpretation | Visualization |
|---|---|---|
| covariate-alter | Effect of my friend’s behavior on friendship | |
| covariate-ego | Effect of my behavior on friendship | |
| same covariate | Birds of a feather flock together |
Three simulations from wave 1 to wave 2 according to M1: edges on y axis ordered by number of occurences in simulation
Simulate Wave 2 from Model 1 1000 times. Show edges that show up more than 5% of the time.
Chose additional parameters because of very low \(p\)-values (\(p\) < 0.001) according to RSiena (Ripley et al 2016)
Some Questions:
Pearson, M. and Michell, L. 2000. “Smoke Rings: Social Network Analysis of Friendship Groups, Smoking, and Drug-Taking.” Drugs: Education, Prevention and Policy 7(1):21-37.
Pearson, M. and West, P. 2003. “Drifting Smoke Rings: Social Network Analysis and Markov Processes in a Longitudinal Study of Friendship Groups and Risk-Taking.” Connections 25(2):59-76.
Ripley, R., Boitmanis, K., Snijders, T.A.B., & Schoenenberger, F. 2016. RSiena: Siena - Simulation Investigation for Empirical Network Analysis. R package version 1.1-304/r304. URL
Steglich, C., Snijders, T.A.B, and Pearson, M. 2010. “Dynamic Networks and Behavior: Separating Selection from Influence.” Sociological Methodology. 40(1):329-393.
Snijders, T.A.B. 1996. “Stochastic actor-oriented models for network change.” Journal of Mathematical Sociology 21:149-172.
Wickham, H., Cook, D., and Hofmann, H. 2015. “Visualizing Statistical Models: Removing the Blindfold.” Statistical Analysis and Data Mining: The ASA Data Science Journal 8(4):204-225.
R Packages Used: tidyverse, geomnet, plotly, tweenr, RSiena, network, sna